A nonlinear analysis on the three oscillation modes of a bifilar suspension pendulum (1) (Formulation of nonlinear oscillation problem and primary resonance analysis of one degree of freedom nonlinear inertia type forced damping systems)

Abstract

A bifilar suspension pendulum, a uniform density bar suspended at its two points by two strings of same length from an upper horizontal plane, may swing in two vertical planes or make torsional oscillation about a vertical axis. The free oscillation periods measured in the three modes match well the normal modes derived from linear theory due to the pendulum configuration. These modes are linearly independent of each other, but it is possible to make nonlinear coupling between those as various types of internal resonance. For each mode, a common equation of inertia type shows to give nonlinear hardening/softening. Swing mode 1 has softening as in the simple pendulum, but Swinging-bar mode 2 makes softening/hardening mainly depending on the configuration. Rotational oscillation mode 3 also makes softening/hardening with changing the moment of inertia and the configuration. These are shown analytically and numerically by methods of singular perturbation and numerical computations, prior to an analysis of the internal resonances in a forthcoming paper.